When in beginning algebra class we were to "simplify" an expression. I
put "simplify" in quotes because sometimes I wondered if we didn't end
out with something more complicated than what we started with. But I
have tried to do the same thing with tacit expressions and I have
really had a lot of trouble. Maybe I just need practice, but I have to
rely on doing it like I did it in algebra class, with explicit x and
y.

Here is my shot at "simplifying" your original statement using tacit
expressions. Looks awkward. Anybody have any suggestions?

([^2)+(2**)+(]^2)          NB. (x^2)+(2*x*y)+(y^2)
([^2)+(2*[*])+(]^2)
([*[)+([*])+([*])+(]*])    NB. (x*x)+(x*y)+(x*y)+(y*y)
(([*[)+([*]))+([*])+(]*])  NB. ((x*x)+(x*y))+((x*y)+(y*y))
([*([+]))+(([+])*])        NB. (x*(x+y))+((x+y)*y)
([*([+]))+(]*([+]))        NB. (x*(x+y))+(y*(x+y))
([+])*([+])                NB. (x+y)*(x+y)
(+)*(+)
+*+
*:@+                       NB. (x+y)^2
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